L8: Analysis of Rate Data

L8-1

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.

A0 AA

0A

C 1 XC P

P1 X

A2

Ar ' kC

Review: Pressure Drop in PBRsA → B -rA = kCA

dXA/dW for an isothermal ideal gas phase reaction with DP1st order reaction rate

Mole balance

Rate law

Stoichiometry (put CA in terms of X)

Combine A0 AA

A0

k C 1 X 1 WdXdW F

0

P 1 WP

Pressure drop (put P/P0 in terms of X) Only for =0

& Isothermal

0

AA

A0dX

Fd

r 'W

Process is like an onion → layer built upon layer& sometimes it makes you cry

L8-2


A0 AA

0A

C 1 XC P

P1 X

A2

Ar ' kC

Review: Pressure Drop in PBRsA → B -rA = kCA

dXA/dW for an isothermal ideal gas phase reaction with DP1st order reaction rate

Mole balance

Rate law

Stoichiometry (put CA in terms of X)

Combine A0 AA

A0

k C 1 X 1 WdXdW F

0

P 1 WP

Pressure drop (put P/P0 in terms of X) Only for =0

& Isothermal

0

AA

A0dX

Fd

r 'W

How do we determine the reaction order?

L8-3


L8: Analysis of Rate DataGoal: how to determine rate laws

In practice, collection and analysis of rate data is the most time consuming task in reactor design

BMBKinetics

StoichiometryFluid dynamics

Reactor volume Reactor design problem

BMBReactor VolumeStoichiometry

Fluid dynamics

KineticsBEFORE

Reactor design problem

L8-4


Review of Rate LawsThe reaction: is elementary and irreversible. Which of the following is true?

a) b) c)

d) The rate cannot be determined from this informatione) None of the above

kC H g C H g H g2 6 2 4 2

2 6 2 4 2C H C H Hr kC C

2 6 2 6C H C H-r kC

2 6 2 6C H C Hr C

Ethanol and acetic acid react to form ethyl acetate and water. The rate of ethyl acetate formation is 1st order in ethanol conc and 0th order in acetic acid conc. Which of the following is true?

a) rethyl acetate = kCethyl acetateCwater

b) rethyl acetate = kCethanolCacetic acid

c) rethyl acetate = kCethanol

d) rethyl acetate = kCacetic acid

e) rethyl acetate = kCethanol2Cacetic acid

-1

Cacetic acid0 (zero power) = 1

L8-5


Collection & Analysis of Rate Data

• Constant-volume batch reactor– For homogenous reactions– Concentration vs. time measurements– Measurement during the unsteady-state operation

• Differential reactor– For solid-fluid reactions– Measurement during steady state operation– Product concentration is usually monitored for different feed conditions

Data collection is done in the lab, where we can simplify BMB, stoichiometry, and fluid dynamic considerationsGoal: determine reaction order, , and specific reaction rate constant, k, in the rate law• Want ideal conditions → well-mixed (data is easiest to interpret)

•Select a simple reactor

L8-6


Method of ExcessA + B → products Suspect rate eq. -rA = kCA

CBb

1.Run reaction with an excess of B so CB ≈ CB0

2.Rate equation simplifies to –rA = k’CA where k’=kACB

b ≈ k’=kACB0b and

can be determined

3.Repeat, but with an excess of A so that CA ≈ CA0

4.With excess A, rate simplifies to –rA = k’’CBb where k’’=kACA

≈ k’’=kACA0

5.Determine kA by measuring –rA at known concentrations of A and B, where

13AA

A B

r 1k dm molsC C

b

b

L8-7


Analysis Methods

• Differential method• Integral method• Half-lives method• Initial rate method• Differential reactor• More complex kinetics

L8-8


Differential Methodj0 j j j

dF F r V Ndt

0 0

jj

dCr

dt Where –rA = kCA

alpha power

b) Determine dCA/dt from plot by graphical or numerical methodsc) Plot ln(-dCA/dt) vs ln CA

Average slope

AA

dCln lnk lnCdt

Slope = A

A

dC dtkC

To find k, find the value of –dCA,p/dt that corresponds to a specific concentration CA,p. Raise CA,p to the power and divide into –dCA/dt)p

Hey, we just jumped from step a to step c. How do we get dCA/dt?

a) Plot DCA/Dt as a function of t

jj j j

dCdr V C V r V Vdt dt

AA

dC kCdt

L8-9


3. dCA/dt is read using the value where the curve crosses a specified time

Graphical Method1. Plot DCA/Dt vs t2. Draw rectangles on the

graph. Then draw a curve so that the area above the curve that is cut off of each rectangle approximately fills the unfilled area under the curve

ACt

D

D

0 t1 t2

DCA/Dt)t=0

DCA/Dt)t=1

DCA/Dt)t=2

At

dCdt

0

At

dCdt

1

At

dCdt

2

L8-10


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt8 0

4 1

2 2

1 3

L8-11


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt8 0 1-0=1

4 1 2-1=1

2 2 1

1 3

L8-12


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt8 0 1 4-8= -4

4 1 1 2-4= -2

2 2 1 1-2= -1

1 3

L8-13


Graphical Method ExampleCA t Dt DCA DCA/Dt8 0 1 -4 4

4 1 1 -2 2

2 2 1 -1 1

1 3

CA t Dt DCA DCA/Dt -dCA/dt8 0 1 -4 4

4 1 1 -2 2

2 2 1 -1 1

1 3

-

L8-14


Graphical Method ExampleCA t Dt DCA DCA/Dt8 0 1 -4 4

4 1 1 -2 2

2 2 1 -1 1

1 3

CA t Dt DCA DCA/Dt -dCA/dt8 0 1 -4 4 4.5

4 1 1 -2 2 2.55

2 2 1 -1 1 1.35

1 3 0.5

-

L8-15


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt8 0 1 -4 4 4.5

4 1 1 -2 2 2.55

2 2 1 -1 1 1.35

1 3 0.5

AA

dCln lnk lnCdt

Slope =

A

A

dC dtkC

-Plot ln(-dCA/dt) vs ln CA

AA

dC kCdt

L8-16


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt ln(-dCA/dt) ln(CA)8 0 1 -4 4 4.5 1.5 2.1

4 1 1 -2 2 2.55 0.9 1.4

2 2 1 -1 1 1.35 0.3 0.7

1 3 0.5 -0.7 0

AA

dCln lnk lnC

dt

Slope =

A

A

dC dtk

C

= 1.0

.k . 1

4 5 0 68

Plot ln(-dCA/dt) vs ln CA

-rA= (0.6/time)CA

L8-17


Graphical Method ExampleCA t Dt DCA DCA/Dt -dCA/dt8 0 1 -4 4 4.5

4 1 1 -2 2 2.55

2 2 1 -1 1 1.35

1 3 0.5

AA

dCln lnk lnCdt

Slope =

A

A

dC dtkC

Differential MethodOnly for irreversible reactions

Advantages: 1 experiment

Disadvantages: can only handle simple kinetics

--

L8-18


Analysing methods

Differential method• Integral method• Half-lives method• Initial rate method• Differential reactor• More complex kinetics

L8-19


Integral Method

• A trial-and-error procedure to find reaction order• Guess the reaction order → integrate the differential

equation• Method is used most often when reaction order is known

and it is desired to evaluate the specific reaction rate constants (k) at different temps to determine the activation energy

• Looking for the appropriate function of concentration corresponding to a particular rate law that is linear with time

L8-20


For the reaction A products

For a first-order reaction - rA = k CAA

AdC kC

dt

A0

A

Cln kt

C

ln (CA0/CA)

t

2AA

dC kCdt

For a second-order reaction - rA = k CA2

A A0

1 1 ktC C

1/CA

t

For a zero-order reaction -rA = k AdC kdt

A A0C C kt

CA

t

AA

dC rdt

Plot of CA vs t is a straight

line

Plot of ln(CA0/CA) vs t is a straight

line

Plot of 1/CA vs t is a straight line

L8-21


Analysis Methods

Differential method Integral method• Half-lives method• Initial rate method• Differential reactor• More complex kinetics

L8-22


Method of Half-lives

• The half-life of a reaction, t1/2, is defined as the time it takes for the concentration of the reactant to fall to half of its initial value

• By determining the half-life of a reaction as a function of the initial concentration, the reaction order and specific reaction rate can be determined

L8-23


Method of Half-lives

• The half-life of a reaction, t1/2, is defined as the time it takes for the concentration of the reactant to fall to half of its initial value

• By determining the half-life of a reaction as a function of the initial concentration, the reaction order and specific reaction rate can be determined

L8-24


AA

dC kCdt

A products 1 1

A A0

1 1 1tk 1 C C

A A0 1 21C C at t = t2

1

1 2 1A0

2 1 1tk 1 C

1

1 2 A02 1ln t ln 1 lnCk 1

ln (t1/2)

ln CA0

Slope = 1-

A Ar kC

Plot ln(t1/2) vs ln CA0. Get a straight line with a slope of 1-α

L8-25


Analysis Methods

Differential method Integral methodHalf-lives method• Initial rate method• Differential reactor• More complex kinetics

L8-26


Method of Initial Rates• When the reaction is reversible, the method of initial rates can

be used to determine the reaction order and the specific rate constant

• Very little product is initially present, so rate of reverse reaction is negligible– A series of experiments is carried out at different initial

concentrations

– Initial rate of reaction is determined for each run

– Initial rate can be found by differentiating the data and extrapolating to zero time

– By various plotting or numerical analysis techniques relating -rA0 to CA0, we can obtain the appropriate rate law:

A0 A0r kC

L8-27


Example: Initial Rate Method

4HCl + CaMg(CO3)2 Mg2+ + Ca2+ + 4Cl-+2CO3 + 2H2O

The dissolution of dolomite using hydrochloric acid:

Concentration of HCl at various times was determined from atomic absorption spectrophotometer measurements of the Ca2+ and Mg2+ ions

CHCl

t

4 N HCl

1 N HCl

A0 A0r kC

Make a plot of ln (-rA0) vs ln CA0 The slope =

L8-28


Evaluating the mole balance on a constant V batch reactor at t = 0:

HClHCl 0 HCl,0

0

dC(r ) kC

dt

HCl

HCl,00

dCln lnk lnC

dt

CHCl, 0

(N)

Initial reaction rate –rHCl,0

(mol/cm2 s) x 107

1 1.2 4 2.0 2 1.36

0.1 0.36 0.5 0.74

ln (CHCl)

ln (-rHCl,0)

Slope =

Plot of ln (-rHCl,0) vs ln CHCl,0 will give reaction

order & k

L8-29


Analysis Methods

Differential method Integral methodHalf-lives method Initial rate method• Differential reactor• More Complex Kinetics

L8-30


Differential Reactors• The criterion for a reactor being differential is that the conversion

of the reactants in the bed is extremely small, as is the change in reactant concentration through the bed

• Reactant concentration through the reactor is essentially constant (i.e. the reactor is considered to be gradient-less)

• Can treat the mole balance like a CSTR• Rate of reaction determined for a specified number of pre-

determined initial or entering reactant concentrations• Determine rate of reaction as a function concentration or partial

pressure• Operate isothermally

CA0 CAeCA

CA0 ~ CA~ CAe

L8-31


Differential Catalyst BedThe rate of reaction per unit mass of catalyst, r’A

flow rate in - flow rate out + rate of generation = rate of accumulation

A0 Ae AF F r W 0 D

A0 Ae 0 A0 AeA

F F C Cr

W W

D D

When constant flow rate, 0 = :

D D0 p0 A0 Ae

ACC C

rW W

Product concentration

The reaction rate is determined by measuring product concentration, Cp

L8-32


More Complex Kinetics

• Carry out batch experiments• Use optimization software to compute kinetic parameters by least squares

(covered in process control)

• Investigate errors by calculating standard deviations of parameters and looking at magnitudes of ra,meas,i –rA, calc,i to look for outliers (will learn in process design, this is FYI for this class

• If parameters are sufficiently accurate, then stop. If not, keep repeating the procedure

# data pts 2

A,measurement i A,calc,ii 1

r r

Sum of squares difference between the measured values and calculated values

L8: Analysis of Rate Data

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